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Fourier transforms of vector measures on some semigroups. (English) Zbl 0942.28019
The authors consider Fourier transforms of vector measures on the semi-group \(G\) defined by a non-void set with a complete order relation. It is made into a semi-group by the operation \(xy=\max (x,y)\). This semi-group is supposed to be compact.
Generalization of analogous results by the same authors are obtained.
The main result is the following: If \(X\) is a Banach space and \(\Gamma \) the semi-group of semicharacters of \(G\), a function from \(\Gamma \) into \(X\) is the Fourier transform of regular \(X\)-valued Borel measure on \(G\) if and only if it is weakly continuous and of bounded variation.
28E10 Fuzzy measure theory
28B05 Vector-valued set functions, measures and integrals
43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.